Some fixed point iteration procedures

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(1)

RESEARCH EXPOSITORY

AND

SURVEY

ARTICLE

SOME

FIXED

POINT ITERATION PROCEDURES

B.E. RHOADES

DepartmentofMathematics IndianaUniversity Bloomington, IN 47405

U.S.A.

(Received August 15, 1989)

ABSTRACT.

This paperprovides a survey of iteration procedures that have been used to

obtainfixedpoints for maps satisfying a variety ofcontractive conditions. The author does not claim toprovide complete coverageofthe literature, and admits to certain biases in the

theoremsthat arecited herein. Inspite oftheseshortcomings, however, this papershould be

auseful reference for those personsvishingtobecome better acquaintedwiththearea.

KEY

WORDS AND PHRASES.fixedpointiteration.

1980 AMS

SUBJECT CLASSIFICATION

CODE. 47H10, 54H25

1.

ITERATION

PROCEDURES.

The literature abounds with papers which establish fixed points for maps satisfying a

variety ofcontractive conditions. Inmostcasesthe contractive definition isstrongenough,not onlytoguaranteethe existence ofauniquefixedpoint, but alsoto obtainthatfixedpoint by repeatediteration of thefunction.

However,

forcertain kinds of maps, suchas nonexpansive

maps, repeatedfunction iterationneed not converge to a fixed point.

A

nonexpansive map satisfiesthecondition

IITx

Tyll <_

]Ix

Yll

foreach pair of points x, yinthespace.

A

simple exampleisthefollowing. Define

T(x)

1-xfor0

_<

x

_<

1. Then

T

isanonexpansive selfmap of

[0,1]

withauniquefixedpointatx

1/2,

but,ifonechooses as astarting point the value

z a,a

1/2,

then repeatediterationofTyields thesequence

{1

-a,a, 1

-a,a,...).

In

1953

W.R.

Mann

[32]

defined the following iteration procedure. Let

A

be a lower triangularmatrix withnonnegativeentriesandrowsums 1. Define xn+l

T(vr,),

where

Vn

E

ankXk"

k=O

The mostinterestingcasesof theManniterativeprocessareobtainedby choosingmatrices

A

such that

a,+l,k

(1-a,+,,+)a,t,,k

0,1,...,n;n 0,1,2,...,andeither a,, 1 for alln ora,,

<

1for alln

>

0. Thus,ifonechoosesany sequence

{c,}

satisfying

(i) co

1,

(ii)

0

_<

c,,

<

1 forn

>

0, and

(iii)

c, c, thenthe entriesof

A

become

atn C/t

ank=ck

H

(1-cjl,

k<n.

(1.1)

j=k+l

(2)

2 B.E. RHOADES

Theabove representation for

A

allowsoneto writetheiterationschemein thefollowing form: x,+,

(1

c,.,)x,.,

+

c,.,

T(

x,

).

One exampleofsuchmatrices is the

Cesaro

matrix,obtainedby choosingc,,

1/(n

+

1).

Anotheris

c.

1 for alln

>_

O, which correspondsto ordinaryfunctioniteration, commonly calledPicard iteration.

Pichard iterationof the function

$1/2

(I

+

T)/2,

is equivalent to theManniteration

schemewith

ank

-This matrix isthe Eulermatrixof order 1,and the transformation

SI/

has beeninvestigated byt’]delstein

[13]

and Krasnoselskii

[30].

Krasnoselskii showed

that,

if

X

isauniformlyconvex

Banach space, and

T

is a nonexpansive selfmapof

X,

then

5’/

converges to a fixed point ofT. Edelsteinshowed that theconditionofuniformconvexity could be weakenedtothatof

strictconvexity.

Pichard iteration of the function

Sx

AI

+

(1

A)T,

0

<

<

1, for any function

T,

homogeneousofdegree 1,isequivalent to theManniteration schemewith

This matrix isthe Eulermatrixoforder

(1

A)/A.

Theiterationof

5’x

has beeninvestigated by Browder andPetryshrn

[7],

Opial

[36],

and Schaefer

[50].

Mannshowed

that,

if

T

is any continuousselfmapofa closedinterval

[a,

b]

withatmost

onefixed point, thenhis iterationscheme,with c,,

1/(n

+

1),

converges to thefixedpoint of T. Franks and

Marzec

[15]

extendedthisresult to continuous functions possessingmorethan

onefixedpointintheinterval.

A

matrix

A

iscalledaweightedmeanmatrixif

A

isalowertriangularmatrix with nonzero entries

an

p/Pn,

where

{p

}

isanonnegative sequence withp0 positiveand

Pn

Z

p cx,

k--0

The author

[42]

extended the above-mentioned result of Franks andMarzec toany

con-tinuousselfmapofaninterval

[a, b],

and

A

anyweightedmeanmatrixsatisfying the condition

lim

]an,

an-l,t O.

k--0

In

[42]

the author also showed that thematrix definedby

(1.1)

isequivalent toaregular weightedmeanmatrix withweights

ckp0

k

>

0.

H(

)’

Let E

be aBanach space,

C

aclosedconvexsubset of

E,

T

acontinuous selfmap of

C.

Mann

[32]

showed

that,

if either of the sequences

{xn}

or

{vn }

converges, then sodoes the

other,

and to thesamelimit, whichis afixedpoint of

T. Dotson

[12],

extendedthisresult to locallyconvexHausdorfflineartopologicalspacesE. Consequently,tousethe

Mann

iterative

(3)

ForuniformlyconvexBanachspaces,the followingwasobtainedindependently byBrowder

[6],

Kirk

[281,

and Gohde

[6].

Let C beaclosed,

bounded,

andconvexsubset ofauniformlyconvex Banachspace,

T

a

nonexpansiveselfmapofC. Then

T

hasafixedpoint.

Unfortunatelythe proofs of above theoremarenotconstructive.

A

number of papers have beenwritten to obtainsomekindof sequential convergence for nonexpansivemaps. Mostsuch theoremsarevalidonlyundersomeadditional hypothesis, suchascompactness,andconverge only weakly. Someexamples oftheorems ofthis type appear in

[8].

Halpern

[18]

obtained two algorithmsfor obtaining fixed points for nonexpansive maps on Hilbert spaces.

In

his

dissertation, Humphreys

[23]

constructed an algorithm which canbeapplied toobtain fixed points for nonexpansivemapsonuniformlyconvexBanachspaces.

A

nonexpansivemappingis saidto be asymptotically regularif, for each point x inthe space,

lim(T"+lx

T"z)

0.

In

1966 Browder and Petryshyn

[7]

established the following result.

THEOREMI.

([7,

Theorem

6]).

Let

X

beaBanachspace,

T

anonexpansive asymptot-icallyregularselfmap ofX.

Suppose

that

T

hasafixedpoint, and that

I- T

mapsbounded closed subsets of

X

intoclosed subsets ofX. Then,for eachz0 E

X, {Tnx0

converges toa

fixedpoint of

T

inX.

In

1972 Groetsch

[17]

established the following theorem, which removes the hypothesis that

T

beasymptoticallyregular.

THEOREM

2.

([17,

Corollary

3]).

Suppose T

isanonexpansive selfmap ofaclosedconvex

subset

E

of

X

whichhas at leastonefixed point. If

I-

T

mapsbounded closedsubsets of

E

intoclosed subsets of

E,

then the Manniterativeprocedure, with

{c,}

satisfyingconditions

(i), (ii),

and

(iv)

c,,(1

c,,)

oo, convergesstronglytoafixedpoint ofT.

Ishikawa

[25]

established the following theorem.

THEOREM

3. Let

D

beaclosed subset ofaBanachspace

X

and let

T

beanonexpansive

mapfrom

D

intoacompact subsetofX. Then

T

hasafixed pointin

D

and theManniterative process with

{c,}

satisfyingconditions

(i)- (iii),

and 0

_<

c,

<

b

<

1for all n, convergestoa

fixedpoint ofT.

For spaces ofdimension higher than one, continuity is not adequate to guarantee

con-vergence toafixed point, either by repeated function iteration, or by some otheriteration

procedure. Therefore it is necessarytoimposesomekindofgrowthconditionon themap. If thecontractivecondition isstrong enough,then themapwillhaveaunique fixed point,which

canbeobtainedby repeatediterationof thefunction. If thecontractive condition isslightly weaker, thensomeotheriterationschemeisrequired. Evenif the fixed pointcanbe obtainedby functioniteration,itisnotwithoutinterestto determine ifotheriteration

procedures

converge tothefixedpoint.

A

generalizationofanonexpansive map with at leastonefixed point that ofa quasi-nonexpansive map.

A

function

T

isaquasi-nonexpansivemapifithas at leastonefixed point,

and,

for eachfixedpointp,

[[Tx

-p[[

_<

[Ix

-p[[.

The followingisdue toDotson

[12].

THEOREM

4. Let

E

beastrictlyconvexBanach space,

C

aclosedconvexsubset of

E,

T

acontinuousquasi-nonexpansiveselfmapofCsuch that

T(C)

C

K

C

C,

where

K

iscompact. Let x0 E

C

and consider a

Mann

iterationprocess such that

{c,}

clusters at somepoint in

(0,1).

Then the sequences

{x,,},

{v,}

convergestronglytoafixed point of

T.

(4)

4 B.E. RHOADES

thespace, atleast oneof the followingistrue:

(i)

IlTx

Tyll

<_ ,llx

yll, (ii)

[]Tx

Ty[]

<_/3[[[x

Tx[[

+

]]y- Ty[[],

or

(iii)

[[Tx

Ty[[

<_

3’[[Ix

Ty[[

+

[[y-

Tx[[],

where a,/3,3’ arereal nonnegative constantssatisfying a

<

1,/3,3’

<

1/2.

Asshownin

[52],

T

has a unique fixedpoint, which can be obtainedby repeatediteration of the function. The following resultappears in

[42].

THEOREM5.

([42,

Theorem

4]).

Let

X

beauniformlyconvexBanachspace,Eaclosed

convexsubset of

X, T

aselfmapof

E

satisfyingconditionZ. Then the

Mann

iterativeprocess

with

{c,}

satisfyingconditions

(i),

(ii),

and

(iv)

convergestothe fixedpoint ofT.

A

generalization of definition

Z

was made by Ciric

[11].

A

mapsatisfies condition Cif

thereexists a constant k satisfying 0

_<

k

<

1 such

that,

for eachpair of points x,y inthe

space,

IlTx

Tyll

<

k

max{llx

ll,

Ilx

TxII,

Ilu

TyII,

II

Tull,

II

In

[42]

theauthorproved thefollowing for Hilbertspaces.

THEOREM6.

([42,

Theorem

7]).

Let

H

beaHilbert space, Taselfmapof

H

satisfying

condition C. Then the Mann iterative process, with

{c,}

satisfying conditions

(i)-(iii)

and

limsupc,

<

1 k2 convergestothe fixed point ofT.

Chidume

[10]

has extended the above result to gv spaces, p

>

2, under the conditions

k2(p-1)<

1 andlimsupc,

<(p-1)--k

.

Asnoted earlier,if

T

iscontinuous,then, iftheManniterativeprocess converges,itmust converge toafixedpoint of

T.

If

T

is not continuous, thereis no

guarantee

that,evenif the

Mann

processconverges,itwill converge toafixed point of

T.

Consider,forexample,themap

T

definedbyTO T1 0,

Tz

1,0

<

z

<

1. Then

T

isaselfmap of

[0,1],

withafixedpoint

at x 0.

However,

theManniterationscheme,with

cn

1/(n

+

1),

0

<

0

<

1, convergesto

1,which isnot afixed point ofT.

A

Inap Tis said tobe strictly-pseudocontractiveifthere existsaconstant k, 0

_<

k

<

1 suchthat,for all pointsx,y in thespace,

[ITx

Tyll

<_

Ilx

y]l

+

kll(I

T)z -(I

T)yII

.

Weshall call denote the class ofall suchmapsby

P2.

Clearly

P2

mappings contain the

nonex-pansivemappings,but the classesP2,

C,

andquasi-nonexpansivemappingsare independent. THEOREM 7.

([42,

Theorem

8]).

Let

H

beaHilbert space,

E

acompact,convexsubset of

H,

T

a

P2

selfmapofE. Then the Mann iteration scheme with

{c, }

satisfying conditions

(i)-(iii)

and limsupc, c

<

1 k,convergesstrongly toafixedpoint ofT.

A

pseudocontractive mappingis a

P2

mappingwith k 1.

Let

P3

denotethe familyof pseudocontractivemappings. Hicks andKubicek

[22]

gave anexaxaple ofa

P3

mappingwith

afixedpoint such that the

Mann

iterativeprocedure, withc,.,

1/(n

+

1),

doesnotconverge tothefixedpoint.

Ishikawa

[24]

defined the followingiterativeprocedure. Given anyx0 in the space, define

x,+,

a,T[nTx,

+

(I

ft,)x,.,]

+

(I

where

{a, }, {/3, }

aresequences of positive numberssatisfyingthe conditions 0

<

c,, _</3.

<_

1,lim/3. 0, c,,13. oo.

He

thenestablished thefollowing result.

THEOREM8. Let

E

beaconvex, compactsubset ofaHilbert space

H,

T

alipschitzian

(5)

Qihou

[38]

extended the above result tolipschitzian hemicontractive maps.

A

hemicon-tractive map is a pseudocontractive map withrespect to afixed point; i.e., if p is any fixed

point of

T,

andx is anypointinthespace,then

T

satisfies

IITx

pll

<_

IIx

p]l

+

]Ix

Tx]]

.

Neitherthe proof of Qihounor Ishikawacanbe used toestablishasimilarresult for the Manniterativetechnique.

In

1968Kannan

[27]

introducedacontractivedefinition which did not require continuity of themap. There then followedaspate offixedpoint papers dealingwithfixed points fora

variety of mapswhich werenot necessarilycontinuous. For example, maps oftype C and

Z

need not becontinuous. Inanearlier paperthe author

[45]

partially ordered manyof these definitions.

As

noted above,it has been shown that theMann iterationscheme converges for

someofthese. Forothercontractive definitionsit isnotpossible todeterminewhetherornot the

Mann

orIshikawaiteration processesconverge.

However,

in some eases it is possible to showthat, iftheydo converge, thentheymust convergetoafixed point ofT. The following theorems illustratethis idea.

THEOREM

9.

([42,

Theorems

5,6]).

Let

X

beaBanachspace,

T

aselfmapof

X,

T

EC or

T

E

P2.

Let

{c,,}

satisfy theconditions

(i), (ii),

andboundedawayfromzero. Then,ifthe corresponding Manniteration processconverges,itconvergestoafixedpointofT.

THEOREM

10.

([46,

Theorem

1]).

Let

X

beaclosedconvexsubset ofanormed space,

T

aselfmapof

X

satisfyingthepropertythat there exist constants c, k

>

0,0

<

k

<

1 such that,for each x,y

X,

IITx

Ty[[

<_

k

max{cllx

ttll, [ll

Txll

/

I[Y

TYH],

Ill

x

TyII

+

Ily

TII]}.

Let

{c,,}

satisfyconditions

(i), (ii),

and limc,

>

0. If theMann iterationscheme converges, thenit convergestoafixedpoint ofT.

A

contractivedefinitionwhich isindependent of that of Ciric isthe following, duetoPal andMaiti

[37].

For

each pair of pointsinthespace,atleastoneof thefollowingconditionsis satisfied:

(i)

[Ix

Tx[[

+

[[y

Ty[[ <_

a[[x

y[[,

1

<

ot

<

2,

(ii)

[Ix

Tx[I

+

I[Y- TYI[

<-

{[[x

Ty]l

+

I[Y- Tzl[

+

x

Y]I},

1/2

_</3

< 2/3,

(iii)

]Ix

Tx]]

+

I]Y

Ty]]

+

]]Tx

Ty]]

<_

7{]Ix

Ty]]

+

]]y

Tx]]},

1

_<

7

<

3/2,

(iv)

IlTx

Tyll

<_

6

max{l]x

Y]I,

[[x

Tx][,

]IY

Ty]],

[]lx

Tyll

+

[[y

Txl[]/2

},

0

<

6

<

1.

Using this demitiontheauthor established thefollowingresult.

TI-ImOIM

ll.

([46,

Theorem

2]).

Let

X

beamanaeh space, aselfmapof

X

satisfying the bove contractive definition. Then if the

Mann

iteration

scheme,

with

{cn}

satisfying conditions

(i), (ii),

andlim

,

>

0, converges,itconvergestoafixed point of

T.

Similar results have been established for the Ishikawa procedure.

In

its original form the Ishikawaprocedure does notincludetheMann iteration processbecauseof the condition 0

<_

cn

_</3n

_<

1.

For,

ifthe Ishikawaprocesswere toincludetheMann processas aspecial case, thenonewould have to assign

each/3n

tobe zero, forcing then each

cn

tobezero.

In

an

effort to haveanIshikwatypeiterationschemewhichdoesinclude theManniterativeprocess

asaspecial case,someauthorshavemodifiedtheinequalityconditiontoreal 0

_<

n,/3n

_<

1. This

change

isreflectedinthefollowingtwotheorems.

THFOREM

12.

([35,

Theorem

1.2]).

Let

X

be anormed linearspace, (7aclosedconvex

(6)

6 B.E. RHOADES

If the Ishikawa iteration scheme, with the

an

bounded away from zero, converges to a

point p,thenp isafixed point ofT.

THEOREM 13.

([40).

Let

E

beaclosedconvexsubset ofaBanachspace,

X, T

aselfmap

of

X

satisfying the condition: thereexists constantsc,k

_>

0,0

_<

k

<

1, such that, for each pair of points z,y in

X,

][Tx

Tyl[

k

max{cllx

ll,

[}lx

Txll

/

Ily

TYl]],[]I

x

TyII

/

Ily-

Txll]).

If theIshikawascheme, with

{an}, {fin}

satisfying theconditions 0

<

a

_<

a,,

_<

1, 0

_<

fl,

_<

1

and limfl, 0, converges toapointp, thenp isafixed point ofT.

In

some theorems ofthistype,the contractivedefinition is weakenoughthat theapriori

existence ofa fixed point for

T

cannot be assured. Thus the convergence of the iteration procedure also yields the existenceofafixed point.

The literature also contains theorems ofthis type for other contractive definitions. In

1988 the author

[47]

proved

that,

formany contractive definitions, eventhoughthe function need not becontinuouseverywhere,it isalwayscontinuousinaneighborhoodofafixed point. Therefore,assuming the convergence of theMann procedureistantamount,insomecases, to

assuming theconvergencetoafixedpoint,inlight of the early result ofMann

[32].

Wenowreturntononexpansivemaps.

THEOREM 14.

([1]).

Let

K

beaclosedboundedconvexsubset ofaHilbertspace

H,

T

a nonexpansiveselfmapofK. Foreach pointeof

K,

theCesarotransform of

{T’e}

converges weaklytoafixed point of

T.

If

T

is also an odd map, then Baillon

[2]

showed that the convergence of the Cesaro transform of

{The}

isstrong. Healsoextendedthe above theorm to

L

spaces.

Wenowmentionsomeresults for othermatrixtransformsof iteratesofT.

THEOREM 15.

([5,

Theorem

1]).

Let

T

beaHilbert space, Caclosed boundedconvex

subset of

H,

T

anonexpansiveselfmapof

C.

Suppose

that

A

isan infinite matrixwith zero

column limits and satisfies lim,,

k(a,,,k+l

a,,k)

+

0. Then, for eachx in

C,

{

a,.,,T’x}

converges weakly toafixedpoint ofT.

A

sequence x is said to be almost convergent to a limit q if

lim,[x,,

+

x,,+l

+...

+

x,.,+,_

]/n

q, uniformly inp.

An

infinite matrix

A

issaid to bestrongly regularif

A

assigns

a limit toeach almost convergent sequence.

Necessary

and sufficient conditions for

A

to be strongly regularwereestablishedbyLorentz

[31].

Theseconditionsarethat

A

be

regular

and alsosatisfy lim,

E

la-,*+

a-l

0.

A

uniformlyconvexBanachspace

X

hasamodulus ofconvexity

$(e)

definedby

(5(e)

ink’{1

]Ix

yll/2

IIll

1,

IIll

1,

I1

yll

},

0

<

2.

Let

{u,,}

beabounded sequence inaclosedconvexsubset CofX. Define

rm(x)

sup{llu.

zll-

n

>_ m},

d denote by c,,, the unique point in C with the property that

r,,,(c,,,)

inf{rm(x)’x

E

C}.

Then

limcm

c, andcis called the asymptotic center of

{u,,}.

(See,

e.g.

[41)

Bruckprovedthefollowingtheorem.

(7)

C, A

astrongly regularmatrix, the A-transform of

T"x

convergesweaklytoafixed pointp, which is the asymptoticcentcr of

{T’x}.

Schoenberg

[51]

obtainedthefollowingcharacterizationfor nonexpansive mapson aHilbert space.

THEOREM

17.

([51]).

Let Tbeanonexpansive ofaHilbertspace

H,A

beanonnegative

matrixwithrowsums oneandzerocolumn limits,

k----0

Then thefollowingareequivalent;

(a) {S,}

isweakly convergent tothe asymptotic centerzof

{T"x}.

(b) {,.,c

isweakly convergent toafixed point for

T.

(c)

IfVisaweak limit point of

{S,},

then

lim,...oo(Re(T"z

T"+*z,z

Ty))

0

(d)

Ifyisaweak limit point of

{S,},

then

(e)

Each weak limit point of

{S,}

is afixedpoint ofT.

Thefollowingtheorem establishesafixed point foracontractive definition which includes

nonexpansivemappings.

THEOREM

18.

[34].

Let

X

be auniformly convex space,

K

a nonempty closed and uniformlyconvexsubset of

X,

T

aselfmapof

K

satisfying

IITx

Tyl[

<

a(x,

y)llx

Yll +

b(x,

y)llx

b’(x,y)lly

Tyll

+

c(x,y)llx

Tyll

+

c’(x,y)lly

Txll

(1.2)

where

a,b,b’,c’

>

O,(a

+

b+

b’

+

c

+

c’)(x,y)

<

1, and

b’(z,y)

b(y,z),c’(x,y)

c(x,y),

for allx,yin

X.

If

sup

(b+c’)(x,y)

<

1

(1.3)

and

inf

IIx

TxlI

0 forsomefiniter then

T

hasafixedpoint in

K.

Define

(1.4)

(y)-

a.lly-

T# ell",

n O, 1,2,...,1

<

p<co.

(1.5)

j=0

where

A

isanynormegative triangularmatrixwithrow sumsone, y, e, E

K. For

eachndefine

s

tobethe unique point of

K

where

(1.5)

assumes its minimum.

THEOREM

19.

([48,

Theorem

2]).

Let

1

<

p

<

oo,

X

e’,K

a nonempty closed,

(8)

8 B. IS. RItOADES

THEOREM 20.

([48,

Theorem

3]).

Let 1

<

p

< o,X

gp,

K

a noneinpty closed, bounded, convex subset of

X,

T a quasi-nonexpansive selfmap of Ix’. If each subsequential weaklimit of

{s

isafixedpoint of

T,

then

{s

converges weakly toafixedpoint ofT.

The specialcaseoftheabovetheoremsin which

A

is the Cesaromatrix and

T

is

nonex-pansive appear in

Beauzamy

andEnflo

[4].

Weshallnowexaminetwootheriterationmethods forobtainingfixedpoints. Thefirstof theseis dueto Kirk

[29]

and is defined asfollows. Let

{x,,}

be any sequence of pointsinthe

space. Then theiterationprocedureisdefinedbythe operator

S

a,T’,

(1.6)

’-0

where kisafixedinteger, k

>_

1,a,

>_

0, 0,1,2,...,a

>

0 and

=0

a, 1.

For

T

any nonexpansive selfmap ofa convexset, S and

T

have the samefixed points.

(See

[29].)

LetTbea selfmapofa Banachspace

X

satisfying, for each x,y in

X,

IIT:

Tull

<_

mx{ll:

ulI,

[11:

T:II

+ Ilu

TulI]/.,

[11:

Tull + Ilu

T:II]/2}.

(1.7)

In

[41]

it is shown that, ifS satisfies

(1.6),

then S and

T

have the samefixed points. The

following result is then proved.

THEOREM 21.

([41,

Theorem

2]).

Let

X

be a uniformly convex Banach space,

K

a

bounded closedconvexsubset of

X,

Taselfmap of

K

which satisfies condition

(7),

Sdefined

by

(6).

If

I-

Smapsbounded closed subsets of

K

intoclosedsets, then,for each x0 E

K

the

sequence

{S"x0 }

convergestoafLxed point of

T

in K.

Theotherfixed point iterationprocedureisdefinedby

s=

a, 1,

aa+

7 0 for at least oneinteger k.

=0

Massa

[33]

has shovnthat,ifTisquasi-nonexpansive, then Sand

T

have thesamefixed

points. Hehas also provedthefollowing.

THEOREM22.

([33,

Theorem

2]).

LetIfbeaclosedconvexsubset ofauniformlyconvex

Banachspace

X,

T

aselfmap of

K

satisfying

IITx

Tull

<_ ,(, u)llx

Yll

+

b(:,

g)(ll:

T:II

+ Ilu

TulI)

+

,:(:,

Y)(II

Tull

+

Ilu

T:II),

where a,

b,

c,

>_

0,

(a

+

2b+

2c)(x,

y)

_<

1,inf,yeK

b(x, V)

>

0,and

T

has atleastonefixed point. Then

{S"x

converges to theonlyfixedpoint ofT.

2.

RATE OF CONVERGENCE.

There hasbeennosystematicstudy of therateof convergencefor these iteration

proce-dures,

andit isdoubtfulif anyglobalstatementscanbemade,sincethereisnothing about these

iteration proceduresto cause theiranalysis to bedifferent fromthat of other approximation methods.

Theauthor

[44]

obtainedsomeevidenceonthebehaviorof the

Mann

iterationprocedure for the decreasing functions

f(x)

1

xrn,g(x)

(1

z)

for 1

_<

rn <: 6 and c,,

(9)

bisectionmethod,accurate to 10places. Then both theManniterationand Newton-Raphson methodswere employed to find each fixed point to within 8 places, using initial guesses of

x0 .1,.2,...,.9. The author

[43]

also examined thefunctions gform 7, 8,...,29, withan

initial guess ofx0 .9,anda,

,

(n

+

1)

-1/2. In

eachcasetheManniterativeprocedure convergedto afixedpoint

(accurate

toeight

places)

in9-12iterations, whereas the Ishikawa

method required from 38 to 42 iterationsfor the same degree ofaccuracy. For increasing

functionsthe Ishikawamethodisbetter than theMannprocess, but ordinary functioniteration isthebest of all three for increasing functions.

An

examinationof the printout showed that theNewton-Raphsonmethod converges faster than the

Mann

scheme. This isnot surprising,since theNewton-Raphsonmethod converges quadratically, whereas the Mann process converges linearly.

However,

whereas the rate of convergenceofthe Newton-Raphson methodis very sensitive to thestarting point,the rate of

convergence for theMannprocessappears to be independent of theinitialguess.

3.

STABILITY.

Weshallnowdiscussthe question of stability ofiteration processes,adopting thedefinition

of stability thatappears in

[19].

Let

X

beaBanach space,

T

aselfmapof

X,

andassumethatxn+l

f(T,x,)

definessome

iterationprocedure involving T. For example,

f(T, Xn)

Tz

Suppose

that

{

x, converges to

afixed pointpofT. Let

{y,}be

anarbitrarysequence in

X

anddefine

forn 0,1,2, Iflim, 0 implies that lim, y, p, then theiteration procedure

f(T,

x)

issaidto beT-stable. ThefirstresultonT-stable mappingswasprovedbyOstrovski

for the Banachcontractionprinciple.

In

[20]

the authors show that functioniteration isstable foravariety ofcontractivedefinitions. Their best result forfunction iteration isthefollowing.

THEOREM

23.

([20,

Theorem

2]).

Let

X

be a complete metric space,

T

a selfmap of

X

satisfying thecontractive condition ofZamfirescu. Let p be the fixed point ofT. Let

xo

_

K,

set x,+l

Txn,n >_

O. Let

{y,}

beasequence in

X,

and set

,

d(y,+l,Ty,)

for

n 0,1, 2, Then

<

+

+

0,

k=O k-O

where

=max

a,l_,l_7

and

lim y, pif andonlyif lim e

n=0.

For

theManniterationproceduretheirbestresultis thefollowing.

THEOREM

24.

([20,

Theorem

3]).

Le

(X,

[[.[[)

beanormed linear space,

T

aselfmapof

X

satisfyingthe Zamfirescu condition.

Let

x0 E

X,

and suppose that thereexistsafixed pointp

and z, p, where

{x,}

denotes the

Mann

iterativeprocedureswiththe

{c,}

satisfying

(i),

(ii),

and0

<

a

<

c.

<

b

<

1.

Suppose {y.}

isasequencein

X

ande.

IIv,.+-[(1-c.)y.+c.Tv.]II

(10)

0 B.E. RHOADES

liP-

Y-+I]

-<

liP-

x-+ll +

(1

a4-

a)n+l

[Ix0

2b(1

12-[-

a)n+l

[Ix,- Tx,[[

+

(1

a

+

=0 =0

where

{

t=max

C,l_fl,

l_7

and

n=0,1,2,...,

lim y,, pif andonlyif lim

,,

0.

Fortheiterationmethod of Kirk

[29],

they have the following result.

THEOREM 25.

([20,

Theorem

4]).

Let

(X,

I1"

II)

be Ba=chspce,

T

seifmp of

X

satisfyingtheBanachcontractivecondition

ilT-Tull

<

cllx-vll

forsomeconstant c,0

<_

c

<

1.

Letp bethe fixedpoint of

T,

z0 an arbitrary point ofX. Set

k

Xn+l

a,T’x,,

forn 0, 1,2,...,

t--0

wherekisaninteger such that k

>_

1,a,

>_

0for 0,1, 2,...,k,al

>

0,and k

e,,

l[,

aiT’Y’ll

forn 0,1,2,....

t-’0

Then

i=0 t=O j---0

n 0,1,2,...

and

lim y,, pif and onlyif lim,, 0.

In

arecent paperthe author

[49]

has

proved

each of the above theorems foracontractive

definitionindependent of that ofZamfirescu.

Consider thefollowingcontractivecondition: thereexistsaconstantcsatisfying0

<

c

<

1

suchthat,for each pair ofpointsx,y in

X,

IITx

Tyll

<

cmax{llx

Vii,

Ilx

TVlI,

IlU

Txll}.

(3.1)

Ourfirstresultisthe following.

THEOREM 26.

([49,

Theorem

1]).

Let

(X, d)

bea completemetricspace,

T

aselfmap of

X

satisfying

(3.1).

Let p be the fixed point ofT.

Let

x0 E

X

and definex,,+l

Tx,. Let

{y,,

}

CX. Define e,-,

d(y,,+

l,

Ty,,).

Then

d(p,

yn+

<

d(p,

Xn+l

+

rrlcn+l-k

d(xk,

Xk+ -1-

crt+l

d(xo,

Yo

"+

Crt-k

k,

(11)

where m

1/(1

c),

and lim,-oo pif andonlyif lim,,_o

,

0.

THEOREM

27.

([49,

Theorem

2]).

Let

(X,

I1"

II)

be a Banachspace,

T

aselfmap of

X

satisfying

(3.1),

pthefixedpoint ofT. Let

{z,}

denote theMann iterativeprocess withthe

{c,}

satisfying

(i)- (iii),

lim

II(1

ci

+

cci

o,

and

[xx

(1

c,

+

cc, converges .=0i=j+!

Let (y,}

C

X

and define

,,

Ily.+

(1

c.)y.

c,

Ty,

ll.

Then

lip-

./,

<

lip-i=0i=j+l

H(1

c,

+

,:,)11o

uoll +

(1

c,

+

cc,)e.,

i=o /=oi=j+

whereit isunderstood that theproductis1 when j n. Then

hm /,,=p ifandonlyif lim e,=0.

Thestabilityresult forKirk iterationshas been extended to thetwoindependent contrac-tivedefinitionsmentioned earlier.

THEOREM

28.

([49,

Theorem

3]).

Let

X

beaBanachspace,

T

aselfmap of

X

satisfying

condition

(3.1).

Let p be the fixed point of

T,

xo

E

X,

and define

{Z,+l}

as in

(1.6).

Let

{!/,,

}

C

X,

anddefine

k

.

Ilu.

,T’u.II

fo. 0, 1,2,

Then,

withm

1/(1- c),

j=0 i=0 i=0

._

(EiCi)n+l

I1 o

o11/

(o, 0

i=0 j=0 i=0

and lim,,_.o. p if andonlyif lim,-ooe, 0.

THEOREM

29.

([49,

Theorem

4]).

Let

X

beaBanach space,

T

aselfmapof

X

satisfying

conditionZ.

Let

p be thefixedpointofT,z0E

X,

and define

{Z,+l}

asin

(1.6).

Let

{/,}

C

X,

anddefine

k

,,

IIv.

,T’u.II

fo., 0, 1, 2,....

(12)

12 B.E. RHOADES

Then, with

{

=max a,

1_,1_

7

+

t=0 ./=0

vand lim,-oo y, p if andonlyiflim,-.ooe, 0.

Weshallnowprove that theIshikawa iterationprocedure is T-stablefor mapssatisfying

either condition

Z

or

(3.1).

THEOREM 30. Let

(X,

I1"

II)

be aBanachspace,

T

aselfmap of

X

satisfying

(3.1).

Let

{x, }

bedefinedbytheIshikawaprocess; i.e.,

Xn+

(1 -a.)x.

+a.Tz.,z.

(1

.)x,,

+

.Tx.,

where 0</,,

<

1,0

<

a

<

or.

<

1

for all n, and the

{a.}

satisfy

lim.

1-Ii"=l(1-

ai

+

cai)

0 and

converges.

Let

{y}

C

X,

anddefine

Then

r-0

,=0 j=i+l

.--I

E

H

(1-

a.i

+

ca:i)i,

t=O

wherem

c/(1

c),

and

lim.-oo

y. pif andonlyif

lim.-oo

.

0.

Proof. Letpbe thefixedpoint ofT. Then

(3.2)

II.+i-vll-<

ll(1

a.)x, +a.Tz,,-vll _< (I-

.)II,,-vll

+

a.llTz.-pll.

IIT,,

Tpll _<

m{ll,,

r’ll,

II,,

TPlI,

II’-

T.II)

II,,

PlI.

(13)

I1=.

pll

I1(1

.).

+

$.Tz.

pll

-<

(1

.)llz.

pll

+

.IIT.

pll

< (1

/.

+

c/.)11.

pll

-<

I1.

pll.

Therefore

[[z.+,

PI[

-<

(1

a.

+

ca.)[Ix.

p[[,

and

{[Izn

p[[}

ismonotone decreasing in

n, and soconverges to a limit d

>

0. If d

>

0, then, taking the limit as n oo, in the above inequality, yields d

_<

(1 -a(1 -c))d <

d, acontradiction. Therefore d 0 and

{z.}

converges top. Using

(3.1),

[Ix.

Tx.[[ <

[Ix.

p[[

+

[[Tp

Tx.[[

< (1

+

c)[[x.

p[[,

and

lim.

[Ix.

Tx.[[

0.

Note

alsothat

x.

-

pimplies

z.

p, andhence

[[z.

Tz.[[

O.

For

any x,yin

X,[[Tx-

Ty[[

<_

m[[x-Tx[[

+

c[[x-

y[[,

wherem

c/(1

-c).

Thus

Ilu.+,

pll

<

I1.+

vii

+

I1(1

.).

+

a,.,Tz.

(1

c.)y.

a.T[(1

/.)y,,

+

Z.Tu.]II

+

.

But

SO

and

(3.2)

followsbyinduction.

An

infinitematrix

A

iscalled multiplicative if theA-limitofaconvergentsequenceisequal tosomeconstantmultipleof thelimit of the sequence. If thematrix haszero columlimits

and finite norm, then the multiplieristhelimit oftherow sums.

Returning to the proof of thetheorem, suppose that

,,

0. Let

A

denote the lower triangularmatrixwithnonzeroentries a,. mc,0,

a.k ma._k

11

(1

trj

+

coL/),

k

<

n. j=k+l

Then,

since

Ilz,,

Tz.[[

O,

[]z.

Tz.[[

O,

d

A

ismultiplicative, therighthdside

of

(3.2)

tends tozero n

,

d p.

Suppo

.

p.

0

e.

fly.+,

(I

a.)y.

a.T[(l

$.)y.

+

D.T.]II

Uy.+,

p]]

+

(1

a.)][y.

p[[

+

a.][T[(l

.)y.

+

.ry.]

rp[]

(14)

14 B.E. RHOADES

THEOREM 31. Let

(X,

II

]l)

bea Banachspace,

T

aselfmap of

X

satisfying condition

Z. Let

{

z,

},

y,

},

,

},

a,,

}, {

,,

be asinTheorem30. Then

n--1

E

2a,_,

H

(I

c

U

+

f%)l]z,-

T:r.,ll

+

2a,,]]z,,

t=0

E

2a,,_,

(1

%

+

6%)1]z,

Tz,[

+

,,+

t=0 j--t+l

,=0

and lim,_y, pif andonlyiflim,-_.oe,, 0.

The proofis similarto that of Theorem1andwill therefore beomitted.

Thefollowingestablishesastability theorem for

Massa’s

iterationprocedure.

THEOREM

32.

([49,

Theorem

5]).

Let

T

satisfy the Banachcontractionprinciple. Let x0E

X,

and defineZn+l

SZ,,

whereSsatisfies

(1.8).

Let

{y,,}

C

X,

and define

Then

,=0 j=0 i=0

and lim,,--.oo y,, pif andonlyif lim,..,oo

,

0.

Weconcludewitharesultonstability for nonexpansivemaps. D. deFigureiredo

[26,

p.

230]

establishedthefollowingresult.

THEOREM

33. Let

H

beaHilbert spaceand C a dosed, bounded, and convex subset

in

H

containing 0. If

T

isany nonexpansiveselfmapof

C,

then,foranyz0 in

C,

the sequence

{z, }

definedby

X

T:

x,,_,n

1,2,...and T,,x

---Tx,

"

converges

strongly

toafixed pointof

T.

THEOREM

34.

([19,

Theorem

2]).

Let

x0E

X,

where

(X,

I1"

II)is

nomedlinear space.

Suppose

thatTisanonexpansive selfmap ofX. Let

,r(.+l)

n

+

1

f(T, xn)

Xn+ *n+l gn, n 0, 1,2,...where

Tn+lx

n+2

Tx.

(15)

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